Charge and heat transport through quantum dots with local and correlated-hopping interactions

The transport properties of junctions composed of a central region tunnel-coupled to external electrodes are frequently studied within the single-impurity Anderson model with Hubbard on-site interaction. In the present work, we supplement the model with an important ingredient, namely, the charge-bond interaction, also known as correlated or assisted hopping. Correlated hopping enters the second-quantized Hamiltonian, written in the Wannier representation, as an off-diagonal many-body term. Using the equation of motion technique, we study the effect of the correlated hopping on the spectral and transport characteristics of a two-terminal quantum dot. Two different Green functions (GFs) appear: one of them describes the spectral properties of the quantum dot, the other the transport properties of the system. The calculation of the transport GF requires the knowledge of the spectral one. We use decoupling procedures similar to those which properly describe the standard Anderson model within the Kondo regime and outside of it. For an arbitrary ratio $x$ between the amplitudes of correlated and single-particle hopping terms, the transport GF fulfils the $x\leftrightarrow 2-x$ symmetry of the model. The average occupation of the dot also obeys this symmetry, albeit the spectral function of the quantum dot, calculated within an analogous decoupling scheme as for the transport GF, does not. We identify the physical reason for this behavior and propose a way to cure it. Since the correlated-hopping term breaks the particle-hole symmetry of the model and modifies all transport characteristics of the system, the detailed knowledge of its influence on measurable characteristics is a prerequisite for its experimental detection. Simple, experimentally feasible methods are proposed.

Figure 1

The transport GF of the model at hand is found to be symmetric with respect to $x=1$ to a very good approximation. In panel (a) we show the symmetry for the imaginary part of the transport GF, calculated for a few values of $x$ (as indicated) and $2-x$ (overlapping dashed curves). The differences between the two curves at particular values of the energy are smaller than $1\%$. Panel (b) shows the detailed behavior for $x=0.1$ vs 1.9, and $x=0.5$ vs 1.5, in the region where the differences are largest. The other parameters are ${\varepsilon }_d=-4$, $U=8$, and $T=0.3$.

Figure 2

The imaginary part of the transport GF vs energy $E$ for a number of $x$ values. For $x=1$ the doubly occupied state is blocked. This is visible as the disappearance of the upper Hubbard band. The other parameters for this particle-hole symmetric model read ${\varepsilon }_d=-4$, $U=8$, and $T=0.3$.

Figure 3

Panel (a) shows the transport density of states as a function of energy $E$ calculated for a set of $x$ values for negative $\delta ={\varepsilon }_d+U/2$, with ${\varepsilon }_d=-8$, $U=8$, and $T=0.3$. Panel (b) shows the similar evolution for $\delta =2>0$ and ${\varepsilon }_d=-2$.

Figure 4

The dot's spectral density of states, $N\left(E\right)=-(1/\pi )\mathrm{Im}{\langle \langle d_{\sigma }|d_{\sigma }^{†}\rangle \rangle }_E$, vs energy for several $x$ values. $N\left(E\right)$ changes with $x$ but is not symmetric, $N(E,x)\ne N(E,2-x)$. The other parameters are ${\varepsilon }_d=-4$, $U=8$, and $T=0.3$. The inset shows the $x$ dependence of the average occupation $\langle n\rangle$ per spin for ${\varepsilon }_d=-2,-4,-8$. For this quantity, the departures from perfect symmetry are very small. To increase their visibility, we plot the dependence $\langle n\rangle$ vs $x$ as solid lines, and the dependence $\langle n\rangle$ vs $(2-x)$ as points of the same color.

Figure 5

The dot's density of states for a number of $x$ values. Note the weak changes of the lower Hubbard band with $x$, and the opposite behavior for the upper one. The other parameters are ${\varepsilon }_d=-4$, $U=8$, and $T=0.3$.

Figure 6

(a) Comparison of the spectral and transport densities of states for $x=0.5$ and ${\varepsilon }_d=-5$, $U=8$, and $T=0.03$ calculated by two methods: “m” stands for the matrix method, while “I” for decoupling I. Panel (b) illustrates the symmetry between these quantities for the particle-hole symmetric system with ${\varepsilon }_d=-4$, $U=8$, and $T=0.3$ calculated by the matrix method. For the full discussion of this figure, see Appendix pp5.

Figure 7

The linear thermopower $S$ (main panel) calculated for ${\varepsilon }_d=-4$, $U=8$, and $T=0.3$ vs $x$. The inset shows the $x$ dependence of the linear conductance $G$. The solid lines indicate the thermopower (conductance) calculated for $0<x<1$, while the points correspond to the values obtained for $0<2-x<1$. The symmetry is nearly perfect in both cases.

Figure 8

Conductance and thermopower $S$ (inset), as well as $L/L_0$ vs $x$ as calculated for ${\varepsilon }_d=-8$, $U=8$, and $T=0.3$.

Figure 9

Symmetry of the transport coefficients with respect to gate bias (characterized by $\delta ={\varepsilon }_d+U/2$) for the model with $x=0$. Panel (a) shows the conductance and panel (b) the thermopower. Other parameters are $U=22$, $T=0.5$.

Figure 10

The effect of $x$ variation on the gate bias dependence (characterized by $\delta ={\varepsilon }_d+U/2$) of the conductivity (a) and the thermopower (b) for the model with $U=22$ at $T=0.5$. Increasing $x$ increases the asymmetry of the conductivity. The upper peak remains essentially intact while the height of the lower one decreases. The inset shows that the decrease of the height is faster than linear.

Figure 11

Dependence of (a) the linear conductance and (b) the thermopower on gate bias, characterized by $\delta ={\varepsilon }_d+U/2$, for $x=0.5$, $U=22$, and $T=0.5$. One observes only a small effect of the anisotropy of the couplings on the lower conductance peak, but large changes of the upper peak, namely, a decrease of its height and an increase of its width when increasing the ratio ${\mathrm{\Gamma }}^R/{\mathrm{\Gamma }}^L$.

Figure 12

(a) Differential conductance $G_d$ vs $x$ for $\delta =-4$, $U=8$, $T=0.3$, and a few values of the voltage $V$ as indicated. Panel (b) shows the dependence of $G_d$ on $\delta ={\varepsilon }_d+U/2$ for $x=0.3$ and the same $U$ and $T$. In both panels the linear conductance is shown by red pluses.